This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures, then construct dual cohomology bases and diffuse them to harmonic 1-forms. Next, we construct bases of holomorphic differentials. We then obtain period matrices by integrating holomorphic differentials along homology bases. We also study the global conformal mapping between genus zero surfaces and spheres, and between general meshes and planes. Our method of computing conformal structures can be applied to tackle fundamental problems in computer aid design and computer graphics, such as geometry classification and identification, and surface global parametrization.
Rigidity question have attracted much interest in the past. In the compact case, we have the famous work of Calabi and Vesentini  and Mostow . Whereas Calabi and Vesentini proved a local version, namely that compact quotients of bounded symmetric domains admit no nontrivial deformations in case the domain is irreducible and of complex dimension at least 2, Mostow proved a global rigidity result, at the expense, however, of working only within the class of quotients of symmetric domains. Mostow's work is based on quasiconformal mappings. A different analytic approach was recently undertaken by Siu . If M is a compact K~ ihler manifold diffeomorphic (or, more generally, homotopically equivalent) to a quotient N of an irreducible bounded symmetric domain, he studied a harmonic homotopy equivalence the existence of which is assured by the theorem of EeUs and Sampson, and demonstrated that
In this paper, we construct complete constant scalar curvature Khler (cscK) metrics on the complement of the zero section in the total space of O ( - 1 ) 2 over O ( - 1 ) 2 , which is biholomorphic to the smooth part of the cone <i>C</i> <sub>0</sub> in O ( - 1 ) 2 defined by equation O ( - 1 ) 2 . On its small resolution and its deformation, we also consider complete cscK metrics and find that if the cscK metrics are homogeneous, then they must be Ricci-flat.
Foundation Compositio Mathematica, 1990, tous droits rservs. Laccs aux archives de la revue Compositio Mathematica(http://http://www. compositio. nl/) implique laccord avec les conditions gnrales dutilisation (http://www. numdam. org/legal. php). Toute utilisation commerciale ou impression systmatique est constitutive dune infraction pnale. Toute copie ou impression de ce fichier doit contenir la prsente mention de copyright.